In this video, we will learn how to write and solve one-step multiplication and division equations in a variety of questions, including word problems. Before starting these questions, we’ll familiarise ourselves with the key vocabulary we will be using. Some of the key vocabulary or definitions we will be using are as follows. Firstly, let’s consider what we mean by a linear equation. These are equations that give a straight line when plotted on a graph. For example, 𝑥 plus two is equal to five. 𝑥 minus seven is equal to three. Two 𝑥 is equal to 10. And 𝑥 divided by three is equal to four.
Note here that the equations contain four different operations: addition, subtraction, multiplication, and division. For the purpose of this video, we will focus on those equations involving multiplication and division. This video will also only deal with one-step equations and not two-step or multiple-step equations that you might see later. A one-step equation is an equation that requires only one calculation to solve it. In this particular video, this will once again only involve multiplication and division.
During this video, we will discuss reciprocal operations or inverse operations. These are operations that can undo a calculation. The reciprocal or inverse of adding is subtracting. The reciprocal of multiplying is dividing, and vice versa. It is this key point that we will focus on in this video. We know that four multiplied by three is equal to 12. We also know that 12 divided by three is equal to four.
Dividing both sides of the first equation by three gives us the second equation. This is because the threes on the left-hand side would cancel as three divided by three is equal to one. And we’re left with four is equal to 12 divided by three. Remember, whatever you do to one side of an equation, you must do to the other. We have to perform the same operation on both sides of the equal sign as this ensures that the equation continues to be balanced. We mentioned at the beginning of the video that a key part will be solving equations. This involves finding the value of 𝑥 or the unknown that makes the equation true.
We will now look at some examples, the first of which involve solving a one-step linear multiplication equation.
Anthony bought five bars of chocolate of the same kind, each of the same price. If 𝑓 is the cost of each chocolate bar, solve five 𝑓 equals 25 to determine 𝑓.
We’re told in the question that Anthony buys five bars of chocolate with the same value. The letter 𝑓 represents the cost of one bar. Therefore, the cost of five bars is equal to five 𝑓. Our equation is equal to 25. Therefore, the total cost of the five bars equals 25. In this question, we’re not given the units. So we don’t need to worry if the 25 is in cents, pence, or any other currency.
We have been asked to solve the equation to determine 𝑓. This is an example of a one-step equation as we only need to carry out one calculation to solve the equation. Remember, when solving an equation, you need to perform the same operation to both sides. This keeps the equation balanced. In this case, we’ll divide both sides of the equation by five. We do this because multiplying by five and dividing by five are reciprocal or inverse operations.
Five 𝑓 divided by five is equal to 𝑓. The fives cancel as five divided by five is equal to one. 25 divided by five is equal to five. This means that the value of 𝑓 in the equation five 𝑓 equals 25 is five. In the context of this question, the cost of each chocolate bar is five units.
We will now look at a second example where we need to write and then solve a one-step equation.
It takes James five times longer than it takes Olivia to get to work. If James’s journey is 70 minutes, write an equation for the time 𝑥 it takes Olivia to get to work. Then solve the equation.
In this question, we need to write an equation and then solve it. We’re told in the question that it takes Olivia 𝑥-minutes to get to work. It takes James five times longer. Five multiplied by 𝑥 is equal to five 𝑥. Therefore, it takes James five 𝑥 minutes to get to work. We’re also told that James’s journey time is actually 70 minutes. This means that we have a linear equation five 𝑥 is equal to 70.
As we have now written the equation, we can move on to the second part of the question, which is to solve it. We know that James’s journey time is 70 minutes. And we need to solve the equation five 𝑥 equals 70 to work out Olivia’s journey time. This is a one-step equation as we only need to carry out one calculation to solve it. Remember, we need to perform the same calculation or operation to both sides of the equation. In this case, we will divide by five. Dividing by five is the inverse or reciprocal of multiplying by five.
Five 𝑥 divided by five is equal to 𝑥 as the fives cancel. 70 divided by five is equal to 14. If we couldn’t work out this calculation mentally, we could use the short division bus stop method. Seven divided by five is equal to one remainder two. And 20 divided by five is equal to four. Therefore, 70 divided by five equals 14. The solution of the equation five 𝑥 equals 70 is 𝑥 equals 14. We can therefore conclude that it takes Olivia 14 minutes to get to work.
We can check this answer by considering James’s time and multiplying five by 14. As 70 divided by five was 14, we know that five multiplied by 14 must be equal to 70. Multiplication and division are the opposite or reciprocal of one another. James’s journey time was 70 minutes, which is five times longer than Olivia’s of 14 minutes.
We will now look at a third example where we will set up an equation involving two variables.
Holly paints one garden chair in 12 minutes. Write an equation for the number of chairs 𝑐 that she could paint in ℎ-hours.
We are told in this question that Holly can paint one chair in 12 minutes. However, we’re interested in the number of chairs that she can paint in ℎ-hours. Let’s firstly consider how many chairs Holly could paint in one hour. We know that 60 minutes is equal to one hour. We need to work out the number of chairs that can be painted in 60 minutes.
We could begin to count up in twelves. Holly would be able to paint two chairs in 24 minutes as 12 plus 12 is 24. Adding another 12 gives us 36 minutes. So she can paint three chairs in this time. Holly could paint four chairs in 48 minutes and five chairs in 60 minutes. You might however have noticed immediately that 12 multiplied by five is equal to 60. Either way, we can see that Holly can paint five chairs in 60 minutes or one hour.
We now need to write an equation for the number of chairs 𝑐 that she can paint in ℎ-hours. If Holly can paint five chairs in one hour, she could paint 10 chairs in two hours, 15 chairs in three hours, and so on. In ℎ-hours, she would be able to paint five ℎ or five multiplied by ℎ chairs. Our equation is therefore 𝑐 is equal to five ℎ. We could then substitute in values for ℎ and 𝑐. We could substitute in values for ℎ to work out the number of chairs painted in a certain number of hours or substitute in a value for 𝑐 to work out the number of hours it would take to paint a certain amount of chairs.
Our next example involves a real-world context involving fractions.
A rectangle’s width is one-sixth of its length. Given that the rectangle’s width is nine inches, determine its length.
We will answer this question by firstly drawing a diagram and then setting up a one-step linear equation. Let’s consider a rectangle with width 𝑊-inches and length 𝐿-inches. We’re told in the question that the width is one-sixth of the length. The word “of” in mathematics means multiply. So 𝑊 is equal to one-sixth multiplied by 𝐿. This in turn can be written as 𝑊 equals one-sixth 𝐿 or 𝐿 divided by six.
In this particular question, we’re told that the rectangle’s width is nine inches. We can substitute this into our equation so that nine is equal to one-sixth 𝐿. This is the same as nine is equal to 𝐿 divided by six. In order to solve this equation, we need to perform the same operation on both sides of the equal sign. In this case, we will multiply by six as multiplying by six is the opposite or inverse of dividing by six.
On the left-hand side, six multiplied by nine or nine multiplied by six is equal to 54. On the right-hand side, the sixes cancel. And we’re just left with 𝐿. As 𝐿 is equal to 54, we can conclude that the length of the rectangle is 54 inches. We can check this answer by working out one-sixth of 54. As this is equal to nine, which was the rectangle’s width, we know that our answer of 54 inches is correct.
We’ll now look at our final example of writing and solving a linear equation.
I take a number and divide it by three and then divide by six. The result is two. What is the number?
We can solve this problem in lots of ways. We will look at setting up a linear equation and also using function machines. We will begin by letting 𝑛 be the number. Our first step is to divide this number by three. We usually write this when dealing with algebra as 𝑛 over three. We then need to divide this answer by six. This is the same as multiplying by one-sixth as dividing by number is the same as multiplying by the reciprocal of that number. Multiplying the numerators and the denominators gives us 𝑛 over 18 or 𝑛 divided by 18. Dividing a number by three and then by six is the same as dividing the number by 18.
We’re told in the question that the result or answer is two. Therefore, 𝑛 divided by 18 equals two. We can solve this equation for 𝑛 by multiplying both sides of the equation by 18. This is because the reciprocal of dividing by 18 is multiplying by 18. And we must perform the same operation to both sides. On the left-hand side, the eighteens cancel, leaving us with 𝑛. Two multiplied by 18 is equal to 36. The number that we started with is therefore 36. We can check this answer by firstly dividing 36 by three. This gives us 12. Dividing 12 by six gives us two. This means that our answer of 36 was correct.
As mentioned at the start, an alternative method here would be to use function machines. We are beginning with an input of 𝑛, dividing by three, and then dividing by six and getting an output of two. The reciprocal or inverse of dividing by six is multiplying by six. Dividing by three has an inverse of multiplying by three. If we know that the output is two, we can calculate the input by firstly multiplying by six and then multiplying by three. Two multiplied by six is equal to 12. 12 multiplied by three is equal to 36. This confirms that our input number 𝑛 was equal to 36.
We will now finish this video by summarising the key points. As well as the definitions and vocabulary that we looked at at the start of this video, we need to remember the following to solve one-step equations. Firstly, we need to identify the reciprocal or inverse operation. For example, to solve four 𝑥 equals eight, we have to divide by four to cancel multiplying by four. This is because multiplication and division are reciprocal operations.
Secondly, we need to ensure that we do the operation to both sides. This ensures that the equation remains balanced. Finally, it is important to check your answer by substitution. The solution to the equation four 𝑥 equals eight is 𝑥 equals two. We can check this by substituting two back into the equation. Four multiplied by two is equal to eight. Remembering these three key points will help us solve any one-step equation involving multiplication and division.